cp's OEIS Frontend

A045991 a(n) = n^3 - n^2.

%I A045991 #183 Apr 12 2025 11:55:25
%S A045991 0,0,4,18,48,100,180,294,448,648,900,1210,1584,2028,2548,3150,3840,
%T A045991 4624,5508,6498,7600,8820,10164,11638,13248,15000,16900,18954,21168,
%U A045991 23548,26100,28830,31744,34848,38148,41650,45360,49284,53428,57798,62400,67240,72324
%N A045991 a(n) = n^3 - n^2.
%C A045991 Number of edges in the line graph of the complete bipartite graph of order 2n, L(K_n,n). - _Roberto E. Martinez II_, Jan 07 2002
%C A045991 Number of edges of the Cartesian product of two complete graphs K_n X K_n. - _Roberto E. Martinez II_, Jan 07 2002
%C A045991 That is, number of edges in the n X n rook graph. - _Eric W. Weisstein_, Jun 20 2017
%C A045991 n such that x^3 + x^2 + n factors over the integers. - _James R. Buddenhagen_, Apr 19 2005
%C A045991 Also the number of triangles in a 2 X n grid of points and therefore also (n choose 2) * (n choose 1) * 2, or (2n choose 3) - 2*(n choose 3). - _Joshua Zucker_, Jan 11 2006
%C A045991 Nonnegative X values of solutions to the equation (X-Y)^3-XY=0. To find Y values: b(n)=(n+1)*n^2 (see A011379). I proved that, if(X,Y) is different from (0,0) and m=2, 4, 6, 8, 10, 12,..., then the equation (X-Y)^m-XY=0,... has no solution. - _Mohamed Bouhamida_, May 10 2006
%C A045991 For n>=1, a(n) is equal to the number of functions f:{1,2,3}->{1,2,...,n} such that for a fixed x in {1,2,3} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and _Milan Janjic_, Mar 13 2007
%C A045991 a(n) equals the coefficient of log(2) in 2F1(n-1,n-1,n+1,-1). - _John M. Campbell_, Jul 16 2011
%C A045991 Define the infinite square array m(n,k) = (n-k)^2 for 1<=k<=n below the diagonal and m(n,k) = (k+n)(k-n) for 1<=n<=k above the diagonal. Then a(n) = Sum_{k=1..n} m(n,k) + Sum_{r=1..n} m(r,n), the "hook sum" of the terms left from m(n,n) and above m(n,n). - _J. M. Bergot_, Aug 16 2013
%C A045991 Partial sums of A049451. - _Bruno Berselli_, Feb 10 2014
%C A045991 Volume of an extruded rectangular brick with side lengths n, n and n-1. - _Luciano Ancora_, Jun 24 2015
%H A045991 Vincenzo Librandi, <a href="/A045991/b045991.txt">Table of n, a(n) for n = 0..1000</a>
%H A045991 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A045991 R. J. Mathar, <a href="/A045991/a045991.pdf">On the Diophantine equation (X-Y)^m-XY=0</a>.
%H A045991 Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H A045991 J.S. Seneschal, <a href="/A045991/a045991.jpg">Oblong cuboid illustration</a>.
%H A045991 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>.
%H A045991 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%H A045991 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A045991 G.f.: 2*x^2*(x+2)/(-1+x)^4 = 6/(-1+x)^4+10/(-1+x)^2+14/(-1+x)^3+2/(-1+x). - _R. J. Mathar_, Nov 19 2007
%F A045991 a(n) = floor(n^5/(n^2+n+1)). - _Gary Detlefs_, Feb 10 2010
%F A045991 a(n) = 4*binomial(n,2) + 6*binomial(n,3). - _Gary Detlefs_, Mar 25 2012
%F A045991 a(n+1) = 2*A006002(n). - _R. J. Mathar_, Oct 31 2012
%F A045991 a(n) = (A000217(n-1)+A000217(n))*(A000217(n-1)-A000217(n-2)). - _J. M. Bergot_, Oct 31 2012
%F A045991 From _Wesley Ivan Hurt_, May 19 2015: (Start)
%F A045991 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).
%F A045991 a(n) = Sum_{k=0..n-1} Sum_{i=n-k-1..n+k-1} i. (End)
%F A045991 Sum_{n>=2} 1/a(n) = 2 - Pi^2/6. - _Daniel Suteu_, Feb 06 2017
%F A045991 Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 + 2*log(2) - 2. - _Amiram Eldar_, Jul 05 2020
%F A045991 E.g.f.: exp(x)*x^2*(2 + x). - _Stefano Spezia_, May 20 2021
%F A045991 Product_{n>=2} (1 - 1/a(n)) = A146485. - _Amiram Eldar_, Nov 22 2022
%F A045991 From _J.S. Seneschal_, Jun 21 2024: (Start)
%F A045991 a(n) = (n-1)*A000290(n).
%F A045991 a(n) = n*A002378(n-1).
%F A045991 a(n) = A011379(n) - A001105(n). (End)
%p A045991 A045991:=n->n^3 - n^2: seq(A045991(n), n=0..50); # _Wesley Ivan Hurt_, Mar 30 2014
%t A045991 Table[n^3 - n^2, {n, 0, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 22 2008 *)
%t A045991 Table[4 Binomial[n, 2] + 6 Binomial[n, 3], {n, 0, 50}] (* _Robert G. Wilson v_, Mar 25 2012 *)
%t A045991 LinearRecurrence[{4, -6, 4, -1}, {0, 4, 18, 48}, 20] (* _Eric W. Weisstein_, Jun 20 2017 *)
%o A045991 (Sage) [n^2*(n-1) for n in range(0, 40)] # _Zerinvary Lajos_, Dec 03 2009
%o A045991 (Magma) [n^3-n^2: n in [0..40]]; // _Vincenzo Librandi_, May 02 2011
%o A045991 (PARI) a(n)=n^2*(n-1) \\ _Charles R Greathouse IV_, Jul 17 2011
%Y A045991 Cf. A011379, A047929, A114364 (essentially the same).
%Y A045991 Cf. A000217, A000290, A000578, A006002, A049451, A146485.
%Y A045991 Cf. A002378, A001105, A011379.
%Y A045991 Cf. A028724, A052289.
%K A045991 nonn,easy
%O A045991 0,3
%A A045991 _N. J. A. Sloane_